Asymptotic distribution of maximum likelihood estimator in generalized linear mixed models with crossed random effects

Generalized linear mixed models (GLMM) with crossed random effects is infamously known to present major challenges not only computationally but also theoretically. In fact, to date only consistency of the maximum likelihood estimators (MLE) has been proved for GLMM with crosses random effects. We introduce a new technique in asymptotic analysis built on a second-order Laplace approximation (LA) of a conditional expectation, whose coefficients are carefully evaluated. The LA leads to a large system of equations involving the conditional expectations, which is asymptotically inverted to obtain explicit approximation to the conditional expectations. This powerful technique leads to a new approach to establishing asymptotic distribution of the MLE when the likelihood function is intractable. As a result, asymptotic normality of the MLE is rigorously established for GLMM with crossed random effects, bringing answer to an open problem dating back to decades ago.